Coherence conversion for optimized resolution in optical measurements – example of femtosecond time resolution using the transverse coherence of 100-picosecond X-rays

Abstract

A way is proposed to obtain a femtosecond time resolution over a picosecond range in laser-pump, X-ray probe spectroscopic measurements where the light source and the detector are much slower than that. It is based on a phase-space transformation from the time/bandwidth to the spatial/wavenumber domain to match the coherence properties of synchrotron radiation to the requirements of femtosecond experiments. In a first step, the geometry of the laser incidence maps time, t, of laser-induced femtosecond dynamics to a spatial coordinate, x. Then, a far-field X-ray diffraction pattern, i.e. the optical Fourier transform, is obtained from the laser-induced modifications of the sample properties, including shifts of X-ray absorption edges and changes in crystallographic unit-cell form factors. Whereas the first step is similar to previously used schemes for femtosecond time resolution, the second one is substantially different with specific advantages discussed in the text. Key to this technique is that the modulus of the Fourier transform is invariant with respect to translations $\mathit{\xi }$ along x, which are due to the $t-x$ correlation. It can, therefore, be acquired in a simple intensity measurement with a slow detector. The phase, which does vary strongly with $\mathit{\xi }$, is missing in the intensity data, but can be recovered through a heterodyning technique. Data from a demonstration experiment are presented. The same concept can be used to obtain attosecond time resolution with an X-ray free-electron laser.

Keywords:

• phase space
• coherence
• laser-pump
• X-ray probe spectroscopy
• Wigner distribution
• femtosecond
• XFEL

Acknowledgements

The experiment sketched in Section 8 was done in collaboration with A DiChiara and EM Dufresne whose help is gratefully acknowledged. I would like to thank the anonymous reviewer for suggesting a consideration of partial coherence.

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 There is still an explicit time dependence in the overall intensities, but not in the angles into which they go. See Figure 3.

2 Angles $\mathit{\delta }$ corresponding to wavenumbers $k=2\mathit{\pi }\mathit{\delta }/\mathit{\lambda }$, where $\mathit{\lambda }$ is the X-ray wavelength.

3 Resolution of the phase ambiguity will require picosecond, but not femtosecond resolution. The geometry of the laser and X-ray illumination can be used to provide a picosecond resolution, see Section 5, so the time resolution of the detector is still irrelevant.

4 The velocity $c/sin\mathit{\chi }$ exceeds the speed of light, but this is purely due to the geometry of wavefronts and does not violate causality.

5 Emittance is the term used in accelerator physics for the phase-space volume occupied by a beam with respect to one particular product of conjugate variables. The longitudinal emittance is the product of particle-bunch length and particle-energy spread, and a transverse emittance (horizontal or vertical) is the product of beam diameter times particle angular spread.

6 It is sufficient to consider scalar amplitudes because vectorial effects, such as polarization mixing, will play no role here.

7 Depending on the details of the laser and X-ray beam geometry, angular projection factors may apply.

8 Both are complex, and at no energy do they become exactly equal to each other.

Additional information

Funding

This work was supported by the US Department of Energy, Office of Basic Energy Sciences [contract number DE-AC02-06CH11357].